Disjoint Convex Bodies Research Articles

Observation routes and external watchman routes

We introduce the Observation Route Problem (ORP) defined as follows: Given a set of n pairwise disjoint obstacles (regions) in the plane, find a shortest tour (route) such that an observer walking along this tour can see (observe) each obstacle from some point of the tour. The observer does not need to see the entire boundary of an obstacle. The tour is not allowed to intersect the interior of any region (i.e., the regions are obstacles and therefore out of bounds). The problem exhibits similarity to both the Traveling Salesman Problem with Neighborhoods (TSPN) and the External Watchman Route Problem (EWRP). We distinguish two variants: the range of visibility is either limited to a bounding rectangle, or unlimited. We obtain the following results:(I) Given a family of n disjoint convex bodies in the plane, computing a shortest observation route does not admit a (clog⁡n)-approximation unless P=NP for an absolute constant c>0. (This holds for both limited and unlimited vision.)(II) Given a family of disjoint convex bodies in the plane, computing a shortest external watchman route is NP-hard. (This holds for both limited and unlimited vision; and even for families of axis-aligned squares.)(III) Given a family of n disjoint fat convex polygons in the plane, an observation tour whose length is at most O(log⁡n) times the optimal can be computed in polynomial time. (This holds for limited vision.)(IV) For every n≥5, there exists a convex polygon with n sides and all angles obtuse such that its perimeter is not a shortest external watchman route. This refutes a conjecture by Absar and Whitesides (2006).

Two extensions of the Erdős–Szekeres problem

According to Suk's breakthrough result on the Erdős–Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n -gon, has at most 2^{n+O\left({n^{2/3}\mathrm log n}\right)} points. We strengthen this theorem in two ways. First, we show that the result generalizes to convexity structures induced by pseudoline arrangements. Second, we improve the error term. A family of n convex bodies in the plane is said to be in convex position if the convex hull of the union of no n-1 of its members contains the remaining one. If any three members are in convex position, we say that the family is in general position . Combining our results with a theorem of Dobbins, Holmsen, and Hubard, we significantly improve the best known upper bounds on the following two functions, introduced by Bisztriczky and Fejes Tóth and by Pach and Tóth, respectively. Let c(n) (and c'(n) ) denote the smallest positive integer N with the property that any family of N pairwise disjoint convex bodies in general position (resp., N convex bodies in general position, any pair of which share at most two boundary points) has an n -membered subfamily in convex position. We show that c(n)\leq c'(n)\leq 2^{n+O\left(\sqrt{n\mathrm log n}\right)} .

On the perimeter of k pairwise disjoint convex bodies contained in a convex set in the plane

We prove the following isoperimetric inequality in R2, conjectured by Glazyrin and Moric. Given a convex body S and k pairwise disjoint convex bodies C1,...,Ck that are contained in S, then Σi=1kPer(Ci)≤Per(S)+2(k--1)Diam(S). Here Per(.) denotes the perimeter of a set and Diam(.) is the diameter of a set.

THE ERDŐS–SZEKERES PROBLEM FOR NON‐CROSSING CONVEX SETS

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies.

LR Characterization of Chirotopes of Finite Planar Families of Pairwise Disjoint Convex bodies

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.

Computing Pseudotriangulations via Branched Coverings

We describe an efficient algorithm to compute a pseudotriangulation of a finite planar family of pairwise disjoint convex bodies presented by its chirotope. The design of the algorithm relies on a deepening of the theory of visibility complexes and on the extension of that theory to the setting of branched coverings. The problem of computing a pseudotriangulation that contains a given set of bitangent line segments is also examined.

On neighbors in geometric permutations

We introduce a new notion of ‘neighbors’ in geometric permutations. We conjecture that the maximum number of neighbors in a set S of n pairwise disjoint convex bodies in R d is O( n), and we prove this conjecture for d=2. We show that if the set of pairs of neighbors in a set S is of size N, then S admits at most O( N d−1 ) geometric permutations. Hence, we obtain an alternative proof of a linear upper bound on the number of geometric permutations for any finite family of pairwise disjoint convex bodies in the plane.

On antipodal and adjoint pairs of points for two convex bodies

The numbers of antipodal and of adjoint pairs of points are estimated for a given pair of disjoint convex bodies inEd.

Helly-type theorems for bodies in the plane with common supports

A theorem of Bisztriczky implies that every pair of disjoint convex bodies in the plane has a common supporting line. In this paper, combinatorial conditions implying the existence of common supporting lines for larger collections of disjoint convex bodies in the plane are exhibited.